8 - Energy conservation in the 3D Euler equations on T2 × R+

Summary

The aim of this paper is to prove energy conservation for the incompressible Euler equations in a domain with boundary. We work in the domain $$\TT^2\times\R_+$$, where the boundary is both flat and has finite measure; in this geometry we do not require any estimates on the pressure, unlike the proof in general bounded domains due to Bardos & Titi (2018). However, first we study the equations on domains without boundary (the whole space $$\R^3$$, the torus $$\mathbb{T}^3$$, and the hybrid space $$\TT^2\times\R$$). We make use of somearguments due to Duchon & Robert (2000) to prove energy conservation under the assumption that $$u\in L^3(0,T;L^3(\R^3))$$ and $${|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\R^3} |u(x+y)-u(x)|^3\,\d x\,\d t=0$$ or $$\int_0^T\int_{\R^3}\int_{\R^3}\frac{|u(x)-u(y)|^3}{|x-y|^{4+\delta}}\,\d x\,\d y\,\d t<\infty,\qquad\delta>0$$, the second of which is equivalent to $$u\in L^3(0,T;W^{\alpha,3}(\R^3))$$, $$\alpha>1/3$$.